Question

The measure of one interior angle of a parallelogram is 30 degrees less than 9 times the measure of another angle. Find the measure of each angle.

Answer 1

Answer:

The interior angles of the parallelogram are either:

XXX

{

159

∘

,

21

∘

,

159

∘

,

21

∘

}

or

XXX

{

3

3

4

∘

,

176

1

4

∘

,

3

1

4

∘

,

176

1

4

∘

}

Explanation:

Case 1: The given relation applies to consecutive angles

Consecutive angles of a parallelogram add up to  

180

∘

Let  

a

and  

b

be consecutive angles (measured in degrees) such that the given relation holds:

XXX

a

=

9

b

−

30

and since  

a

+

b

=

180

→

b

=

180

−

a

XXX

a

=

9

(

180

−

a

)

−

30

=

1620

−

9

a

−

30

=

1590

−

9

a

XXX

10

a

=

1590

XXX

a

=

159

and

XXX

b

=

180

−

a

=

21

Case 2: The given relation applies to opposite angles

Opposite angles of a parallelogram are equal.

Let  

a

be the measure (in degrees) of the referenced opposite angles (and  

b

be the measure of the other two angles:  

a

+

b

=

180

)

We are told

XXX

a

=

9

a

−

30

Therefore

XXX

−

8

a

=

−

30

XXX

a

=

3

3

4

XXX

b

=

180

−

a

=

176

1

4

Answer link

EZ as pi

Sep 25, 2016

The angles are  

21

°

,

159

°

,

21

°

,

159

°

Explanation:

In a parallelogram, there are only two sizes of angle - two are acute and two are obtuse.

Opposite angles are equal .

Consecutive angles are supplementary because they are co-interior angle on parallel lines.

An additional solution to the one given by Alan P is using the fact that the sum of the interior angles of a parallelogram is 360°.

Let one angle be  

x

The size of the other angle is given by  

9

x

−

30

2

x

+

2

(

9

x

−

30

)

=

360

   

←

sum of the angles is 360°

2

x

+

18

x

−

60

=

360

20

x

=

420

x

=

21

°

The obtuse angles are each  

180

−

21

=

159

°

The other option where the given angle is the opposite angle is described is well explained by Alan P.

Step-by-step explanation: